We study the set $D(M,N)$ of all possible mapping degrees from $M$ to $N$ when $M$ and $N$ are quasitoric $4$-manifolds. In some of the cases, we completely describe this set. Our results rely on Theorems proved by Duan and Wang and the sets of integers obtained are interesting from the number theoretical point of view, for example those representable as the sum of two squares $D(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2},\mathbb{C}P^{2})$ or the sum of three squares $D(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2}\sharp \mathbb{C}P^{2},\mathbb{C}P^{2})$. In addition to the general results about the mapping degrees between quasitoric 4-manifolds, we establish connections between Duan and Wang’s approach, quadratic forms, number theory and lattices.

Every finite group $G$ acts on some nonorientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of $G$. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain group presentations which allow one to find the actions realizing the symmetric crosscap number of groups of each group of order less than or equal to 63.

We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in \mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\mathbf{A}$ and $\mathbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space $\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for $\mathbb{H}^{n+1}$ and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in $\mathbb{H}^{n+1}$.

We analyze domination properties and factorization of operators in Banach spaces through subspaces of $L^{1}$-spaces. Using vector measure integration and extending classical arguments based on scalar integral bounds, we provide characterizations of operators factoring through subspaces of $L^{1}$-spaces of finite measures. Some special cases involving positivity and compactness of the operators are considered.

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$, where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$. We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$. We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$. Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.

Using variational methods and depending on a parameter $\unicode[STIX]{x1D706}$ we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$:

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}=0$

$\unicode[STIX]{x1D6FE}=1$

The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

In this paper, we generalize monotone operators, their resolvents and the proximal point algorithm to complete CAT(0) spaces. We study some properties of monotone operators and their resolvents. We show that the sequence generated by the inexact proximal point algorithm $\unicode[STIX]{x1D6E5}$-converges to a zero of the monotone operator in complete CAT(0) spaces. A strong convergence (convergence in metric) result is also presented. Finally, we consider two important special cases of monotone operators and we prove that they satisfy the range condition (see Section 4 for the definition), which guarantees the existence of the sequence generated by the proximal point algorithm.

We study the problem of extending a state on an abelian $C^{\ast }$-subalgebra to a tracial state on the ambient $C^{\ast }$-algebra. We propose an approach that is well suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^{\ast }$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^{\ast }$-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz–Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provide connections to related finiteness questions in graph $C^{\ast }$-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings $R$ with identity. The classical ring of quotients $Q$ of $R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums $\oplus Q$ one has to take the full subcategory of $Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which $Q$ has projective dimension $1$ (Theorem 6.4).

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

Ott, Tomforde and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea, we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets, our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any $M$-step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.

The generating graph $\unicode[STIX]{x1D6E4}(H)$ of a finite group $H$ is the graph defined on the elements of $H$, with an edge between two vertices if and only if they generate $H$. We show that if $H$ is a sufficiently large simple group with $\unicode[STIX]{x1D6E4}(G)\cong \unicode[STIX]{x1D6E4}(H)$ for a finite group $G$, then $G\cong H$. We also prove that the generating graph of a symmetric group determines the group.

Let $(X,d,\unicode[STIX]{x1D707})$ be a metric measure space endowed with a distance $d$ and a nonnegative, Borel, doubling measure $\unicode[STIX]{x1D707}$. Let $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup $e^{-tL}$ satisfies a Gaussian upper bound. In this paper, we prove that for any $p\in (0,\infty )$ and $w\in A_{\infty }$, the weighted Hardy space $H_{L,S,w}^{p}(X)$ associated with $L$ in terms of the Lusin (area) function and the weighted Hardy space $H_{L,G,w}^{p}(X)$ associated with $L$ in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duong et al. [‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’, J. Geom. Anal.26 (2016), 1617–1646], who proved that $H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$ for $p\in (0,1]$ and $w\in A_{\infty }$ by imposing an extra assumption of a Moser-type boundedness condition on $L$. Our result is new even in the unweighted setting, that is, when $w\equiv 1$.

In this article, we extend the notion of FP-injective modules to that of Cartan–Eilenberg complexes. We show that a complex $C$ is Cartan–Eilenberg FP-injective if and only if $C$ and $\text{Z}(C)$ are complexes consisting of FP-injective modules over right coherent rings. As an application, coherent rings are characterized in various ways, using Cartan–Eilenberg FP-injective and Cartan–Eilenberg flat complexes.

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that

$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$

$T(V,W)$

$T(V,W)$

$Q$

$W$

$V$

$Q$

$W$

$n$

$m$

$V$

$F$